ÐÏࡱá>þÿ ¦¨þÿÿÿ¤¥çÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿì¥Á}€ ø¿ÿŸbjbj™5™5 �7û_û_†(F*ÿÿÿÿÿÿ·öö99aaaTÿÿÿÿµµµ˜MÔ! µ\Dª-Vƒ:½½½¢".Ð"ä" ÏCÑCÑCÑCÑCÑCÑCF¢¨HdÑCEa¿&!Š¢"¿&¿&ÑC99½½[D‡/‡/‡/¿&‚9Ô½a½ÏC‡/¿&ÏC‡/‡/Æß@� T“B½ÿÿÿÿ€>«ãɵA'öoA»C,D0\DƒA I7)B I(“B Ia“B(ð"ʺ#�‡/J$t¾$ð"ð"ð"ÑCÑCy-ð"ð"ð"\D¿&¿&¿&¿&ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ Ið"ð"ð"ð"ð"ð"ð"ð"ð"ö ÿ:    This packet has been designed to help you review various mathematical topics that will be necessary for your success in Geometry. Instructions: DO ALL PROBLEMS WITHOUT A CALCULATOR. Show all work that leads you to each solution on separate sheets of paper. Additional copies of this packet may be obtained from the Northwest High School website  HYPERLINK "https://northwesths.net" https://northwesths.net ALL work should be completed and ready to turn in on the FIRST DAY of school. This packet will count as part of your first quarter grade. Due Date: Monday, August 31st, 2009 Deadline : Friday, September 4th, 2009 If you have questions regarding the summer math packet, please call Vic Weinstein at (301) 601-4651 ENJOY YOUR SUMMER!! WE ARE LOOKING FORWARD TO SEEING YOU IN THE FALL. Table of Contents Page Objective Suggested Completion Date 4 Fractions ……………………………………. June 28th 5/6 Order of Operations ………………………… July 5th 7 Adding and Subtracting Signed Numbers …. July 12th 8 Multiplying and Dividing Signed Numbers … July 19th 9 Rounding Numbers ………………………. July 27th 10/11 Evaluating Expressions …………………… August 3rd 12 Combining Like Terms …………………… August 10th 13 Proportions ………………………………… August 17th 14 Solving Equations …………………………. August 24th Fractions Simplify the following fractions: 1.  EMBED Equation.3  = 2.  EMBED Equation.3 = 3.  EMBED Equation.3  = Write the following mixed numbers as improper fractions: 4.  EMBED Equation.3  = 5.  EMBED Equation.3  = 6.  EMBED Equation.3  = Perform the indicated operation, and simplify if necessary: 4.  EMBED Equation.3  = 5.  EMBED Equation.3 = 6.  EMBED Equation.3 = 7.  EMBED Equation.3  = 7.  EMBED Equation.3  = 8.  EMBED Equation.3  = 9.  EMBED Equation.3  = 10.  EMBED Equation.3  = 11.  EMBED Equation.3  = 12.  EMBED Equation.3  = 13.  EMBED Equation.3  = 14.  EMBED Equation.3  = 15.  EMBED Equation.3  = 16.  EMBED Equation.3  = 17.  EMBED Equation.3  = Order of Operations To avoid having different results for the same problem, mathematicians have agreed on an order of operations when simplifying expressions that contain multiple operations. 1. Perform any operation(s) inside grouping symbols. (Parentheses, brackets above or below a fraction bar) 2. Simplify any term with exponents. 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. One easy way to remember the order of operations process is to remember the acronym PEMDAS or the old saying, “Please Excuse My Dear Aunt Sally.” P - Perform operations in grouping symbols E - Simplify exponents M - Perform multiplication and division in order from left to right D A - Perform addition and subtraction in order from left to right S Example 1 Example 2 2 – 32 + (6 + 3 x 2) -7 + 4 + (23 – 8 ÷ -4) 2 – 32 + (6 + 6) -7 + 4 + ( 8 – 8 ÷ -4) 2 – 32 + 12 -7 + 4 + ( 8 - -2) 2 – 9 + 12 -7 + 4 + 10 -7 + 12 -3 + 10 = 5 = 7 Evaluate each expression. Remember your order of operations process (PEMDAS). 1. 6 + 4 – 2 " 3 = 2. (-2) " 3 + 5  7 = 3. 15 ÷ 3 " 5  4 = 4. 29  3 " 9 + 4 = 5. 20  7 " 4 = 6. 4 " 9  9 + 7 = 7. 50  (17 + 8) = 8. (12  4) ÷ 8 = 9. 12 " 5 + 6 ÷ 6 = 10. 18  42 + 7 = 11. 3(2 + 7)  9 " 7 = 12. 3 + 8 " 22  4 = 13. 16 ÷ 2 " 5 " 3 ÷ 6 = 14. 12 ÷ 3 - 6 " 2  8 ÷ 4 = 15. 10 " (3  62) + 8 ÷ 2 = 16. 6.9  3.2 " (10 ÷ 5) = 17. 32 ÷ [16 ÷ (8 ÷ 2)] = 18. [10 + (2 " 8)] ÷ 2 = 19. 180 ÷ [2 + (12 ÷ 3)] = 20. ¼(3 " 8) + 2 " (-12) = 21. 5 + [30  (8  1)2] = 22. 3[10  (27 ÷ 9)] = 11 - 22 4  7 23. 5(14  39 ÷ 3) + 4 " 1/4 = 24. [8 " 2  (3 + 9)] + [8  2 " 3] = 25. 162 ÷ [6(7  4)2] ÷ 3 = Operations with Signed Numbers Adding and Subtracting Signed Numbers Adding Signed Numbers Like SignsDifferent Signs Add the numbers & carry the sign Subtract the numbers & carry the sign of the larger number ( + ) + ( + ) = + ( +3 ) + ( +4 ) = +7  ( + ) + (– ) = ? ( +3 ) + (–2 ) = +1 ( – ) + (– ) = – (– 2 ) + (– 3 ) = ( – 5 ) ( – ) + ( + ) = ? ( –5 ) + ( + 3 ) = –2  Subtracting Signed Numbers Don’t subtract! Change the problem to addition and change the sign of the second number. Then use the addition rules. ( +9 ) – ( +12 ) = ( +9 ) + ( – 12) ( +4 ) – (–3 ) = ( +4 ) + ( +3 )  ( – 5 ) – ( +3 ) = ( – 5 ) + ( – 3 ) ( –1 ) – (– 5 ) = ( –1 ) + (+5)  Simplify. Do not use a calculator for this section. 9 + -4 = 7. 20 – - 6 = -8 + 7 = 8. 7 – 10 = -14 – 6 = 9. -6 – -7 = -30 + -9 = 10. 5 – 9 = 5. 14 – 20 = 11. -8 – 7 = 6. -2 + 11 = 12. 1 – -12 = Multiplying and Dividing Signed Numbers If the signs are the same, If the signs are different, the answer is positive the answer is negative Like SignsDifferent Signs (+ ) ( + ) = + ( +3 ) ( +4 ) = +12  ( + ) ( – ) = – ( +2 ) ( – 3 ) = – 6 (– ) (– ) = + ( – 5 ) ( – 3 ) = + 15 ( – ) ( + ) = – ( –7 ) ( +1 ) = –7  (+ ) / ( + ) = + ( +3 ) / ( +4 ) = +12  ( + ) / ( – ) = – ( +2 ) / ( – 3 ) = – 6 (+ ) / ( + ) = + ( +3 ) / ( +4 ) = +12  ( – ) / ( + ) = – ( –7 ) / ( +1 ) = –7  Simplify. Do not use a calculator for this section. (-5)(-3) = 7. -7 = -1 -6 = 8. (3) (-4) = 2 (2)(4) = 9. 8 = -4 -12 = 10. (-2)(7) = -4 (-1)(-5) = 11. -20 = -1 6. -16 = 12. (2)(-5) = 8 Rounding Numbers Step 1: Underline the place value in which you want to round. Step 2: Look at the number to the right of that place value you want to round. Step 3: If the number to the right of the place value you want to round is less than 5, keep the number the same and drop all other numbers. If the number to the right of the place value you want to round is 5 or more, round up and drop the rest of the numbers. Example: Round the following numbers to the tenths place.  Tenths 23.1246 23.1  2. 64.2685 64.3   3. 83.9721 84  Round the following numbers to the tenths place. 1. 18.6231 _____________ 6. 0.2658 ______________ 2. 25.0543 _____________ 7. 100.9158 _____________ 3. 3.9215 _____________ 8. 19.9816 ______________ 4. 36.9913 _____________ 9. 17.1083 ______________ 5. 15.9199 _____________ 10. 0. 6701 ______________ Evaluating Expressions Example Evaluate the following expression when x = 5 Rewrite the expression substituting 5 for the x and simplify. 5x = 5(5)= 25 -2x = -2(5) = -10 x + 25 = 5 + 25 = 30 5x - 15 = 5(5) – 15 = 25 – 15 = 10 3x + 4 = 3(5) + 4 = 19 Evaluate each expression given that: x = 5 y = -4 z = 6 1. 3x 5. y + 4 2. 2x2 6. 5z – 6 3. 3x2 + y 7. xy + z 4. 2 (x + z) – y 8. 2x + 3y – z Evaluate each expression given that: x = 5 y = -4 z = 6 9. 5x – (y + 2z) 13. 5z + (y – x) 10.  EMBED Equation.3  14. 2x2 + 3 11. x2 + y2 + z2 15. 4x + 2y – z 12. 2x( y + z) 16.  EMBED Equation.3  Combining Like Terms What is a term? The parts of an algebraic expression that are separated by an addition or subtraction sign are called terms. The expression 4x + 2y – 3 has 3 terms. What are like terms? Terms with the same variable factors are called like terms. 2n and 3n are like terms, but 4x and 3y are not like terms because their variable factors x and y are different. To simplify an expression, you must combine the like terms.  Simplify 1. 5x + 8x 2. 3y – 6y 5x + 8x = (5 + 8)x = 13x 3y – 6y = (3 – 6)y = -3y 3. 3x + 4 – 2x + 3 4. 2b + 5c + 3b – 6c 3x – 2x + 4 + 3 = (3 – 2)x + 4 + 3 = x + 7 2b + 3b + 5c – 6c = (2+3)b + (5 – 6)c = 5b – c Practice: Simplify each expression 6n + 5n 2. 25b + 15b 37z + 4z 4. x – 5x 3n + 1 – 2n + 8 6. 4f + 5f – 6 + 8 7t + 9 – 4t + 3 8. 2k + 4 – 8k – 1 9. 4r + 3r +6y – 2y 10. 8g + 9h – 4g – 5h 11. 2m + 3n – 4m + 5n 12. a + 5b – 2a + 9b Proportions Solve each proportion for the missing value: 1.  EMBED Equation.3  2.  EMBED Equation.3  3.  EMBED Equation.3  4.  EMBED Equation.3  5.  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(Give your answer in minutes ) 9. Matt paid $33.41 for 13 gallons of gasoline. How many gallons can he buy if he only has $14 ? Solving Equations Solve the following equations. 1. 7x – 17 = 60 2. 5y – 13 = 37 3.  EMBED Equation.3  4. 3(x+2) = 18 5. -2 + 10x = 8x -1 6. 2(a – 3) + 5 = 3(a – 1) 7.  EMBED Equation.3  8. -3(x + 5) = 8x + 18 9. -4y + 3y – 8 = 24 10.  EMBED Equation.3  11. -4r + 5 – 6r = -32 12. 6x + (-3) = -12 13. 6y – 14 – 3y = 8(7 – (-2)) 14. 4c + 5c – 8c = 13 + 6     -  PAGE 1 - Page  PAGE 14 of  NUMPAGES 14 Northwest High School’s Geometry Summer Math Packet DUE THE FIRST DAY OF SCHOOL Student Name: ____________________________________ 2 is less than 5 so keep the 1 the same 6 is greater than 5 so add one to the 2 1 83.9721 + 1 84.0 7 is greater than 5 so add one to the 9 Examples: ²1œYœZœhœÈœÉœËœÍœÐœâœãœ��(�)�1�2�E�F�G�H�Ÿ� �³�´�µ�¶�¿�Ò�×�ö�÷� ž ž ž žBžDžGžýùõñùñùíùèãÜèØÔØÌؽ°ÌØÌØ¡”ÌØ�Ø�ˆ�ylˆ�hdh>¿h³jÝLh³h×V]EHèÿUjôÙöI h×V]CJUVaJjh×V]Uh×V]j^Jhlm„hlm„EHèÿUjPÒöI hlm„CJUVaJjýGhlm„hlm„EHèÿUj\ÑöI hlm„CJUVaJjhlm„Uhzx.hlm„ hã ôhã ô hã ô5� hzx.5�h¿L£hÒ^hå[ hXzxU'\œ]œ^œ_œ±œÉœÊœËœÌœÍœÎœÏœÐœâœãœ��)�*�+�,�-�^�_�`�a�b�—�úúúúúúúúúúúúòòíòèèèèèèèèèèègdzx.gdã ô$a$gdzx.gd—�˜�™�š�›�Ó�Ô�Õ�Ö�×�žžžžžCžDžEžFžGžHžIž‹žŒž�žŽž�ž‘ž“žúúúúúúúúúúúúúúõõõõõõõõõõõóóógd³gdzx.GžIžKžhžqžŽž�ž‘ž’ž”ž•ž—ž˜žšž�žžž¤ž¥ž¦ž§ž«ž²ž³ž¹žºž¼ž½žÁžÂžÌžÍžÏžÐžÓžÔžÝžãžöž÷ž 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